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Putnam 2014 Problems

A1. Prove that every nonzero coefficient of the Taylor series of {(1-x+x^2)e^x} about {x=0} is a rational number whose numerator (in lowest terms) is either {1} or a prime number.

A2. Let {A} be the {n\times n} matrix whose entry in the {i}-th row and {j}-th column is

\displaystyle \frac1{\min(i,j)}

for {1\le i,j\le n.} Compute {\det(A).}

A3. Let {a_0=5/2} and {a_k=a_{k-1}^2-2} for {k\ge 1.} Compute

\displaystyle \prod_{k=0}^{\infty}\left(1-\frac1{a_k}\right)

in closed form.

A4. Suppose {X} is a random variable that takes on only nonnegative integer values, with {E[X]=1,} {E[X^2]=2,} and {E[X^3]=5.} (Here {E[Y]} denotes the expectation of the random variable {Y.}) Determine the smallest possible value of the probability of the event {X=0.}

A5. Let {P_n(x)=1+2x+3x^2+\cdots+nx^{n-1}.} Prove that the polynomials {P_j(x)} and {P_k(x)} are relatively prime for all positive integers {j} and {k} with {j\ne k.}

A6. Let {n} be a positive integer. What is the largest {k} for which there exist {n\times n} matrices {M_1,\dots,M_k} and {N_1,\dots,N_k} with real entries such that for all {i} and {j,} the matrix product {M_iN_j} has a zero entry somewhere on its diagonal if and only if {i\ne j?}

B1. A base {1} over-expansion of a positive integer {N} is an expression of the form {N=d_k10^k+d_{k-1}10^{k-1}+\cdots+d_0 10^0} with {d_k\ne 0} and {d_i\in\{0,1,2,\dots,10\}} for all {i.} For instance, the integer {N=10} has two base 10 over-expansions: {10=10\cdot 10^0} and the usual base 10 expansion {10=1\cdot 10^1+0\cdot 10^0.} Which positive integers have a unique base 10 over-expansion?

B2. Suppose that {f} is a function on the interval {[1,3]} such that {-1\le f(x)\le 1} for all {x} and {\displaystyle \int_1^3f(x)\,dx=0.} How large can {\displaystyle\int_1^3\frac{f(x)}x\,dx} be?

B3. Let {A} be an {m\times n} matrix with rational entries. Suppose that there are at least {m+n} distinct prime numbers among the absolute values of the entries of {A.} Show that the rank of {A} is at least {2.}

B4. Show that for each positive integer {n,} all the roots of the polynomial

\displaystyle \sum_{k=0}^n 2^{k(n-k)}x^k

are real numbers.

B5. In the 75th Annual Putnam Games, participants compete at mathematical games. Patniss and Keeta play a game in which they take turns choosing an element from the group of invertible {n\times n} matrices with entries in the field {\mathbb{Z}/p\mathbb{Z}} of integers modulo {p,} where {n} is a fixed positive integer and {p} is a fixed prime number. The rules of the game are:

(1) A player cannot choose an element that has been chosen by either player on any previous turn.

(2) A player can only choose an element that commutes with all previously chosen elements.

(3) A player who cannot choose an element on his/her turn loses the game.

Patniss takes the first turn. Which player has a winning strategy?

B6. Let {f:[0,1]\rightarrow\mathbb{R}} be a function for which there exists a constant {K>0} such that {|f(x)-f(y)|\le K|x-y|} for all {x,y\in [0,1].} Suppose also that for each rational number {r\in [0,1],} there exist integers {a} and {b} such that {f(r)=a+br.} Prove that there exist finitely many intervals {I_1,\dots,I_n} such that {f} is a linear function on each {I_i} and {[0,1]=\bigcup_{i=1}^nI_i.}

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